A054906 Smallest number x such that sigma(x+2n) = sigma(x)+2n (first definition).

3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 7, 5, 3, 5, 3, 7, 5, 3, 13, 7, 5, 3, 5, 3, 3, 5, 3, 3, 5, 3, 19, 13, 11, 13, 7, 5, 3, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 7, 5, 3, 7, 5, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 5, 3, 3, 13

Offset: 1

Author

Labos Elemer, May 23 2000

Keywords

• nonn

Comments

Least (prime) solutions for phi(x+2n)=phi(x)+2n seems to be identical to this sequence, while prime solutions are indeed identical to this sequence.

2nd definition = smallest number x such that phi(x+2n)=phi(x)+2n.

3rd definition = smallest primes p such that p+2n=q prime (A020483).

The 3 definitions are identical or conjectured to be identical.

The definitions are not identical if we do not take the smallest numbers. These smallest solutions are believed to be always prime numbers.

Duplicate of A020483, assuming that the 3rd definition is also correct. - R. J. Mathar, Apr 26 2015

If it can be proved that all these definitions are identical, then this entry should be merged with A020483. - N. J. A. Sloane, Feb 06 2017

Examples

			n-th primes 2,3,5,7,11,13, are solutions to sigma(x+2n)=2n+sigma(x) at 2n=2,6,22,116,88.
		

References

• Sivaramakrishnan,R.(1989):Classical Theory of Arithmetical Functions. Marcel Dekker,Inc., New York.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A023200-A023203, A015913-A015917, A000203, A000010, A020483.

Programs

Maple

A054906 := proc(n)
    local x;
    for x from 0 do
        if numtheory[sigma](x+2*n) = numtheory[sigma](x)+2*n then
            return x;
        end if;
    end do:
end proc:
seq(A054906(n),n=1..40); # R. J. Mathar, Sep 23 2016

Mathematica

Table[x = 1; While[DivisorSigma[1, x + 2 n] != DivisorSigma[1, x] + 2 n, x++]; x, {n, 100}] (* Michael De Vlieger, Feb 05 2017 *)

PARI

a(n) = my(x = 1); while(sigma(x+2*n) != sigma(x)+2*n, x++); x; \\ Michel Marcus, Dec 17 2013

Formula

Minimal solutions to A000203(x+2n)=A000203(x)+2n or to A000010(x+2n)=A000010(x)+2n or to p+2n=q; p, q primes, a(n)=p.

a(n) <= A054905(n). - R. J. Mathar, Apr 28 2015